Gear Train Torque Equations

In our last blog we mathematically linked the driving and driven gear Force vectors to arrive at a single common vector F, known as the resultant Force vector. This simplification allows us to achieve common ground between F and the two Distance vectors of our driving and driven gears, represented as D1 and D2. We can then use this commonality to develop individual torque equations for both gears in the train.

In this illustration we clearly see that the Force vector, F, is at a 90º angle to the two Distance vectors, D1 and D2. Let’s see why this angular relationship between them is crucial to the development of torque calculations.

First a review of the basic torque formula, presented in a previous blog,

Torque = Distance× Force × sin(ϴ)

By inserting D1, F, and ϴ = 90º into this formula we arrive at the torque calculation, T1 , for the driving gear in our gear train:

T1 = D1× F × sin(90º)

From a previous blog in this series we know that sin(90º) = 1, so it becomes,

T1 = D1 × F

By inserting D2, F, and ϴ = 90º into the torque formula, we arrive at the torque calculation, T2 , for the driven gear:

T2 = D2 × F × sin(90º)

T2 = D2 × F × 1

T2 = D2 × F

Next week we’ll combine these two equations relative to F, the common link between them, and obtain a single equation equating the torques and pitch circle radii of the driving and driven gears in the gear train.